Now, we know that , and thus the dimension of our space of invariants is the dimension of the space. We’ve seen that this is the multiplicity of the trivial representation in , which we’ve also seen is the inner product . We calculate:

This may not be as straghtforward and generic a result as the last one, but it’s at least easily calculated for any given pair of modules and .

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.